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Mandatory course for the master study branch Mathematical analysis. Recommended for the first year of master
studies. Devoted to advanced topics in theory of ordinary differential equations. Content: dynamical systems;
Poincaré-Bendixson theory; Carathéodory theory; optimal control, Pontryagin maximum principle; bifurcation;
stable, unstable and central manifolds.
Last update: T_KMA (02.05.2013)
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Credit is given for active participation in exercises (computing problems at the blackboard) or solving given set of problems at home.
Character of the condition does not allow repetitions. Credit is a neccessary condition for registration for the exam.
The exam consists of a written computative part (90 minutes, three problems, 10 points each) and a following theoretical oral part (30 points). It is neccessary to gain 50% of points in each of the two parts. The exam examines the knowledge of the subject matter in the scope given by the lectures. Last update: Bárta Tomáš, doc. RNDr., Ph.D. (29.09.2017)
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Kurzweil, Jaroslav Ordinary differential equations. Introduction to the theory of ordinary differential equations in the real domain. Translated from the Czech by Michal Basch. Studies in Applied Mechanics, 13. Elsevier Scientific Publishing Co., Amsterdam, 1986. I.I. Vrabie: Differential equations: an introduction to basic concepts, results, and applications, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. H. Amann: Ordinary differential equations, an introduction to nonlinear analysis, de Gruyter Studies in Mathematics 13, Walter de Gruyter & Co., Berlin, 1990. J. Hale, H. Kocak: Dynamics and Bifurcations. Texts in Applied Mathematics 3, Springer, New York, 1991.
Last update: T_KMA (02.05.2013)
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The exam consists of written and oral part. Neccessary condition to take part in the oral part is successful passing the written part. Neccessary condition to take part in the written part is obtaining the credit. If a student does not succeed in the written part, the exam is graded by 'F'. If a student does not succeed in the oral part, both parts of the exam must be repeated at the next attempt. The final grade depends on the points obtained in the written and oral part of the exam
Written part consists of three problems, their topics correspond to the sylabus of the lecture and to the topics of the seminar.
Requirements for the oral part correspond to the sylabus of the course in the extend presented in the lectures. Last update: Bárta Tomáš, doc. RNDr., Ph.D. (28.10.2019)
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1. Dynamical system. Orbit, stationary point, invariant set. Alpha- and omega-limit sets and their properties. La Salle invariance principle. Conjugate dynamical systems. Lemma on rectifications. Poincaré-Bendixson theory in the plane. Bendixson-Dulac criterion of non-existence of periodic solutions.
2. Carathéodory theory - notion of an absolutely continuous solution, local existence and uniqueness.
3. Optimal control theory.
4. Bifurcations. Basic types of bifurcations. Sufficient conditions for existence of bifurcations. Hopf bifurcation.
5. Stable, unstable and central manifolds. Invariance principle and its reformulations. Existence of the central manifold. Approximation of the central manifold. Reduced stability principle. Hartman-Grobman theorem. Last update: T_KMA (16.09.2013)
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Theory of ordinary differential equations (as covered by NMMA333). Last update: Pražák Dalibor, doc. RNDr., Ph.D. (08.05.2018)
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